Sapiens Test

3 min read

The Monty Hall problem: why pigeons win and people don’t

In 2010 six pigeons beat a group of university students at a probability puzzle that once had nearly a thousand people with doctorates writing angry letters to a magazine. The puzzle takes ten seconds to state.

The puzzle

You are on a game show. There are three closed doors. Behind one is a car, behind the other two are goats. You pick door 1. The host, who knows where the car is, opens door 3 and shows you a goat. Then he asks: do you want to stay with door 1 or switch to door 2?

Most people say it doesn’t matter. Two doors, one car, fifty-fifty. The right answer is to switch. Switching wins the car two times out of three.

The letters

The statistician Steve Selvin described the problem in 1975 in two letters to The American Statistician and named it after Monty Hall, the host of the TV show “Let’s Make a Deal”. It became famous in September 1990, when a reader sent it to Marilyn vos Savant’s column “Ask Marilyn” in Parade magazine. She answered: switch.

By her estimate the magazine received about 10,000 letters on the subject, close to 1,000 of them signed by people with PhDs, and most of them said she was wrong. Paul Erdős, one of the most prolific mathematicians in history, reportedly stayed unconvinced until a colleague showed him a computer simulation.

Why switching wins

Forget the open door for a moment and look at your first pick. It is right one time in three. That never changes, whatever the host does afterwards.

So staying wins one time in three and switching wins two times in three. The host’s choice isn’t random. Every time he skips a door, he gives you information.

It is easier to feel with more doors. Imagine 100 of them. You pick one. The host, who knows where the car is, opens 98 others, all goats, and leaves only door 57 closed. Would you still stay with your first pick?

One detail matters. If the host didn’t know where the car was, opened a door at random and it just happened to show a goat, the odds really would be fifty-fifty. The puzzle works because he knows.

The pigeons

In 2010 Walter Herbranson and Julia Schroeder from Whitman College gave the same choice to six pigeons. Instead of doors there were three lit keys, and instead of a car there was grain. A pigeon pecked one key, one of the empty keys went dark, and the bird could peck its first key again or switch to the other lit one.

On the first day the pigeons switched on 36% of trials. By day 30 they switched on 96%.

The researchers ran the same setup with university students, 200 trials each. In their last 50 trials the students switched on about 66% of them.

Why the birds won

The pigeons weren’t doing probability. They were keeping score. Switching paid off twice as often as staying, so they switched more and more until they did it almost every time.

People reason instead, and the reasoning goes wrong in a predictable way. Two closed doors look like fifty-fifty. Staying also feels safer: losing after a switch hurts more than losing with your first choice, so people stick with it. In one study only 13% of 228 participants switched.

The students’ 66% is a trap of its own. Switching on two out of three trials is called probability matching, and it wins less often than switching every time: about 56% of trials instead of 67%. The students learned that switching was usually better, but not that it was always better.

Sources